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  <h1 class="w3-xxxlarge">Group Explorer 3.0 Help: Multiplication Table Overview</h1>
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                <h3>Contents</h3>
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                    <li><a style="text-decoration: none;"
                          href="#introduction-to-multiplication-tables">Introduction to multiplication tables</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#what-multiplication-tables-do-well">What multiplication tables do well</a></li>
                
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            <h2 id="introduction-to-multiplication-tables">Introduction to multiplication tables</h2>
<p>The multiplication table is probably the most widely used tactic for
depicting a group operation.  Let&rsquo;s take a look at some of its potential.
If you have not yet encountered the multiplication table
<a href="../rf-geterms/#visualizers">visualizer</a>, start with these help pages:</p>
<ul>
<li><a href="../rf-groupterms/#multtable">The definition of a multiplication table</a></li>
<li><a href="../rf-um-mt-options/">The documentation on the multiplication table interface in
   <em>Group Explorer</em></a></li>
</ul>
<p>CITE(VGT-4.3 MM-1.5 DE-5.3)</p>
<h2 id="what-multiplication-tables-do-well">What multiplication tables do well</h2>
<p>Coloring the backgrounds of multiplication tables&rsquo; cells reveals patterns in
the group operation.  Let&rsquo;s illustrate this by contrasting
a bad example with a good one.</p>
<h3 id="bad-example">Bad example</h3>
<p><img alt="Multiplication table with unhelpful highlighting" src="../d_4_multtable_bad_example.png" /></p>
<p>The backgrounds of the cells in this multiplication table are colored based
on whether they are in a particular 2-element subgroup. This does not look
as helpful in a multiplication table as it does in a <a href="../rf-groupterms/#cycle-graph">cycle
graph</a> or <a href="../rf-groupterms/#cayley-diagrams">Cayley
diagram</a>, because each element appears
multiple times in a multiplication table.</p>
<h3 id="good-example">Good example</h3>
<p><img alt="Multiplication table with helpful highlighting" src="../d_4_multtable_good_example.png" /></p>
<p>The backgrounds of the cells in this multiplication table are colored based
on their membership in the <a href="../rf-groupterms/#cosets">cosets</a> of the
<a href="../rf-groupterms/#subgroup">subgroup</a> highlighted in red. This shows how the
group operation in this example respects the cosets nicely&ndash;in fact, by
considering each block of like-colored neighbors as one big cell, one can
see the multiplication table of the <a href="../rf-groupterms/#quotient">quotient
group</a>! The subgroup in question is therefore
<a href="../rf-groupterms/#normal-subgroup">normal</a>.</p>
<h3 id="exercise">Exercise</h3>
<p>Consider it an exercise to find a group with a non-normal subgroup and
highlight a multiplication table of that group by the cosets of the
non-normal subgroup.</p>
<p>There are some illustrations can be made from just about any kind of
visualization, including multiplication tables.  For instance, we can use
multiplication tables to illustrate <a href="../s_3_multtable_lattice.png">a lattice of
subgroups</a> or <a href="../s_3_multtable_ses.png">a short exact sequence of
groups</a>, or we could use another kind of
visualization as well.  Both of these advanced uses of multiplication tables
tie in homomorphisms, which are visible on <a href="../rf-um-sheetwindow/">sheets</a>.</p>
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